Finiteness — the series · Crystallography & Symmetry

Why the seventeen is a number at all. The classification is finite because three counts in a row are finite, and the first two are where the work is. Finitely many lattice types, because a lattice's symmetry group is a finite group of integer matrices; finitely many such groups, by Minkowski's lemma and his bound; and finitely many ways to attach translations to each, which is the extension problem. Every step is a count this site makes elsewhere — five, thirteen, seventeen — and this is the reason each of those searches was allowed to stop.

Why there is a list at all

Five lattices, seventeen groups, thirty-two classes, two hundred and thirty. Every one of those counts came out of a search that had to know when to stop, and the reason it could stop is a divisibility Minkowski proved in 1887.

part 1 · restriction

Modulo 3 injective on all thirteen, modulo 2 on 5. Minkowski's lemma says the kernel of reduction modulo an integer of at least three is torsion-free, so a finite group of integer matrices is carried faithfully into a finite group of matrices over ℤ/3 — which is why the classification is finite, before any bound is computed. The middle column checks it on every finite subgroup of GL(2,ℤ) there is: thirteen classes, no collapses. The right column is the case the lemma has to exclude. Modulo 2, minus the identity is the identity, and 8 classes lose operations.

Reduction modulo three

A finite group of integer matrices survives being reduced modulo three: no two of its operations collide. That single fact proves the classification finite without computing any bound — and modulo two it is false, refuted by the inversion centre.

part 2 · restriction

Averaging a metric over the group. The 3 pale ellipses are the unit circle carried by each element of a finite group of rational matrices — none of them a rotation, because the group has been skewed out of the orthogonal ones on purpose. Their average is the heavy ellipse, and it is invariant: MᵀAM = A for every element, exactly, in rational arithmetic. So a finite group of matrices is always a group of isometries of some inner product, and every question about how large such a group can be becomes a question about the symmetries of an ellipse. The space of invariant forms here is 1-dimensional, so up to scale the average is the only one.

The average that makes it finite

Two arguments every classification leans on are usually assumed rather than made: that a finite group of motions fixes a point, and that a finite group of integer matrices preserves a metric. They are the same trick — average over the group — and the trick fails exactly where it should.

part 3 · restriction

The same group means the same pattern

Seventeen patterns is not the same statement as seventeen groups. Two patterns that look nothing alike could in principle have symmetry groups that are abstractly the same, and then the classification would be a classification of drawings. Bieberbach's theorem says they cannot — and the affine map that proves it can be recovered from the two operation sets alone.

part 4 · restriction

Two turns and their undoing leave a slide. A turn g by 90° about the point c and a turn h by 60° about d. The marked point p is carried back 60° about d, back 90° about c, forward 60° about d and forward 90° about c, and does not return: it arrives displaced by a vector of length 2.371, which is 4·sin 45°·sin 30°·|c − d|. Two other points put through the same four motions move by the same vector, drawn beside them, because the commutator g h g⁻¹ h⁻¹ of two rotations of the plane is a translation — (I − A)(I − B)(c − d) exactly — whatever the angles and the centres.

What forces a lattice

Every enumeration here starts from a lattice of translations, and the lattice is usually taken as given. It need not be. A group of motions that is discrete, and leaves no point far from an orbit, has to contain one — in the plane by an argument four lines long, each line a picture, and in space by an inequality whose threshold turns out to be the six-fold rotation.

part 5 · restriction

Three conditions, and a near-miss for each. Zassenhaus's characterisation asks a group for a normal subgroup that is free abelian of finite rank, of finite index, and maximal among the group's abelian subgroups. Four groups against those three clauses. The free group on two letters has no non-trivial abelian normal subgroup at all; the discrete Heisenberg group has one that is free abelian of rank two and maximal abelian, and its index is infinite; ℤ² × ℤ/2 has a free abelian normal subgroup of index two, and the maximal one has torsion in it. Each fails a different clause, which is what shows no clause is redundant. The infinite dihedral group passes and is crystallographic in one dimension.

Which groups a crystal could have

Bieberbach's theorem is a statement about a group acting: discrete, no point far from an orbit. Zassenhaus turned it round into a statement a group can satisfy on its own — a maximal abelian normal subgroup, free of finite rank, of finite index — and each of those three clauses is kept out of redundancy by a group that fails it and nothing else.

part 6 · restriction

An orbit on a parabola, discrete and cocompact. The images of the origin under the group generated by two commuting affine maps of the plane: A slides one step along x and lifts y by the x it started at plus a half, and B is the translation by one in y. The images are the points with whole-number first coordinate and second coordinate a whole number above half the square of it, so the large dots lie on the dashed parabola and the small ones are the rest of the orbit. No two distinct images come closer than 1.000, and no point of the square between the axes lies farther than 0.610 from one — so the action is discrete and its quotient is compact, which is exactly what Bieberbach's first theorem asks for.

Straight lines, and no distances

Every finiteness met so far rests on the motions preserving a metric, because the trick that produces one is an average and an average needs something to average over. Keep the straight lines and drop the distances, and Bieberbach's first theorem is false in the plane — by an example two lines long, whose group is the plane's own translations and whose translations have rank one.

part 7 · restriction

Two, seventeen, two hundred and thirty, and then. The number of arithmetic crystal classes and the number of crystallographic groups in each of the first six dimensions, with the second divided by the first. The classes multiply by between five and fourteen a dimension; the groups multiply by much more, and the quotient — how many groups an average class carries — goes 1.00, 1.31, 3.15, 6.74, 36.5 and 339. The last column says what is derived on this page and what is quoted: the plane in full, six of the seventy-three classes in space, and nothing at all above three dimensions, where the counts come from machine enumerations of the 1970s onwards.

Finitely many is not few

Bieberbach's third theorem says each dimension holds finitely many crystallographic groups and gives no idea how many. The counts are 2, 17, 230, 4783, 222018 and 28927922, and dividing them by the number of arithmetic classes says which of the classification's three steps supplies the explosion — the step that attaches translations, not the one that finds the matrix groups.

part 8 · restriction

How the space groups are spread over their classes. For each number of groups, how many arithmetic classes carry exactly that many space-group types, in space and in the plane, every count derived from the classes' cohomology and the normaliser's action on it. In the plane ten of the thirteen classes carry one group. In space twelve of the seventy-three carry one, thirty-five carry two and fifteen carry four, so the typical class carries two; the long tail is the orthorhombic and tetragonal classes with eight, ten and sixteen.

Two hundred and nineteen over seventy-three

The space groups average three to an arithmetic class, and an average of three could be a few classes carrying dozens or every class carrying a handful. Derived class by class from the cohomology and its normaliser, the answer is the second: twelve classes carry one group, the median carries two, and the largest carries sixteen. Sixty-one of the seventy-three admit a non-symmorphic group, and the step that merges relabellings, which the plane barely uses, removes more than a quarter of the list.

part 9 · restriction

Six matrices and eight: a monoclinic normaliser modulo two. For P2/m and C2/m, every 3 × 3 matrix over the field of two elements that preserves the reductions of the two eigen-sublattices of the class's two-fold or mirror — six on the primitive lattice and eight on the centred one — each drawn as a tile of nine squares filled where the entry is one. The whole infinite normaliser reduces into this set, and a dot under each tile marks that some normaliser element with entries −1, 0 and 1 reduces to it. Every tile is realised, and the same holds in all six monoclinic classes.

The normaliser only acts modulo two

The space groups were counted as orbits of a small, finite set of changes of basis — only those with entries −1, 0 and 1 — out of normalisers that are infinite in the monoclinic classes. That the counts came out right was evidence, not proof. It becomes proof in two steps. A missing change of basis can only leave the count too high, never too low, so the published totals already force every class. And in the monoclinic classes the infinite normaliser acts on the cohomology only through its reduction modulo two, where the small set already reaches everything.

part 10 · restriction

Every finite normaliser of a lattice's group, found whole. The fourteen Bravais lattices on primitive bases, with the number of primitive vectors along the axes of their rotations, the order of the lattice's group, and the order of its normaliser in the integer matrices, found by trying every assignment of axis vectors to three independent ones. On the three lattices up to monoclinic the axis vectors lie on one line or none and the normaliser is infinite. On the other eleven it is finite (48, 16, 48, 48, 16, 16, 24, 12, 48, 48, 48), its largest entry is one, and it is exactly the set of matrices with entries −1, 0 and 1 that the space-group census used.

A normaliser must permute the axes

The space groups were counted with a small set of changes of basis, and past the monoclinic system nothing had shown that set was the whole normaliser. It is, and the reason is geometric: a change of basis that carries a lattice's group to itself must carry each rotation axis to a rotation axis, so it permutes a finite set of vectors, and past the monoclinic system three of them pin it down. Every such permutation tried, the eleven finite normalisers turn out to be exactly the matrices with entries −1, 0 and 1, and 219 and 230 stand without any table behind them.

part 11 · restriction